On the twistor bundle of De Sitter space of S^3_1

EDUARDO HULETT

BEITRAGE R ALGEBRA GEOM

Año: 2008 vol. 49 p. 107 - 123

0138-4821

We study the twistor bundle $cal{Z}$ over De Sitter space $DS$. Viewing $cal{Z}$  as an $SO(1,1)$-principal bundle over the Grassmannian $G_2(b{L}^4)$ of oriented space-like planes in Lorentz-Minkowski $4$-space, the orthogonal complement of the fibers of $pi´: cal{Z} o G_2(b{L}^4) $ defines  a    $4$-dimensional horizontalneutral (of signature $(++--)$) distribution $cal{H} subset Tcal{Z}$. Two $SO(3,1)$-invariant almost Cauchy-Riemann structures $ cal{J}^I$ and $ cal{J}^{II}$ on $cal{H}$ are introduced.  According to which structure is considered two classes of horizontal holomorphic maps arise. These maps are  projected  to $DS$ onto  space-like surfaces with different properties. We characterize both classes of horizontal maps  in terms of the geometry of their projections to $DS$.